IEE5328 Nanodevice Transport Theory and Computational Tools
Prof. Ming-Jer ChenDept. Electronics EngineeringNational Chiao-Tung UniversityMay 1, 2013
Lecture 7:
Effective Mobility in 2DEG and 2DHG of Long-Channel Thick Gate-Oxide Planar Bulk MOSFETs: Microscopic Calculation
(Advanced Device Physics with emphasis onhands-on calculations)
1IEE5328 Prof. MJ Chen NCTU
Two-Dimensional Hole Gas
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Thick Oxides
No Stress
Stress
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National Chiao-Tung University Nano Electronics Physics Lab. 5
Kubo-Greenwood Formula
vxμ is the group velocity of subband μ along x-direction and f0 is the
equilibrium Fermi distribution.
The mobility formula in electron case
is no longer valid for the hole case due to the failure of the effective mass approximation.Thus, the hole mobility formula as derived from the Boltzmann transport equation (BTE) must be used:
National Chiao-Tung University Nano Electronics Physics Lab. 6
Kubo-Greenwood Formula
Group Velocity (cm/sec)
k<100>
(cm-1)
k<010> (
cm
-1)
-4 -2 0 2 4
x 107
-4
-3
-2
-1
0
1
2
3
4x 10
7
-1
-0.5
0
0.5
1
x 108
k<100>
(1/cm)
k<
01
0> (
1/c
m)
-4 -2 0 2 4
x 107
-4
-3
-2
-1
0
1
2
3
4x 10
7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1st SubbandSi (001) @ 300K FS=1MV/cm
National Chiao-Tung University Nano Electronics Physics Lab. 7
Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness
Scattering
; ;
ω
q
Longitudinal Transverse
Acoustic
Optical
Si Phonon Dispersion
H’Hole-AC Phonon
H’Hole-OP PhononΔE≈1meV within 1/2 Brillouin zone
ΔE≈61.2meV
Phonon wave vector
National Chiao-Tung University Nano Electronics Physics Lab. 8
Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness
Scattering
The acoustic deformation potential Dac, is strongly connected to Bir-Pikus deformation potentials. According to Lawaetz, Dac can be formulated as
; ;
c11, c12, and c44 are the elastic coefficients
H’Hole-AC Phonon
Small vibration termElastic scattering
Isotropic approximation
National Chiao-Tung University Nano Electronics Physics Lab. 9
Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness
Scattering
According to Wiley and Costato and Reggiani, the optical deformation potential Dop can have the following formalism:
; ;
Average sound velocity
H’Hole-OP Phonon
Small vibration termInelastic scattering (61.2meV)
Isotropic approximation
ωop : optical phonon frequency; nop : Bose occupation factor of optical phonons
National Chiao-Tung University Nano Electronics Physics Lab. 10
Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness
Scattering
; ;
H’Hole-SR
Small vibration term
Elastic scattering
p+ p+
Gate
Inversion Layern-type Substrate
H’Hole-SR
Anisotropic scattering
National Chiao-Tung University Nano Electronics Physics Lab. 11
Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness
Scattering
; ;
Anisotropic surface roughness scattering rate
0.25 0.30 0.35 0.400
2x1014
4x1014
6x1014
8x1014
Unstressed
Surf
ace
Roughnes
s Sca
tter
ing R
ate
(sec
-1)
Energy (eV)
0o from k<100>
45o from k<100>
135o from k<100>
Long. -1 GPa
(001) 1st SubbandE
eff ~ 0.6 MV/cm
Unscreened
0.05 0.10 0.15 0.20 0.250
1x1014
2x1014
3x1014
4x1014
Unstressed
Long. -1 GPa
(110) 1st Subband E
eff ~ 0.6 MV/cm
Unscreened
Surf
ace
Roughnes
s Sca
tter
ing R
ate
(sec
-1)
Energy (eV)
0o from k<110>
45o from k<110>
90o from k<110>
National Chiao-Tung University Nano Electronics Physics Lab. 12
Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness
Scattering
; ;
Isotropic SR scattering
0
50
100
150
200
250
0
45
90
135
180
225
270
315
0
50
100
150
200
250
(110) x'= <001>y'= <110>
Hole
Mobili
ty (cm
2 /Vse
c)
(001)------- (110) [Pham.] This work
(001) x'= <100>y'= <010>
0
50
100
150
200
250
0
45
90
135
180
225
270
315
0
50
100
150
200
250
(110) x'= <001>y'= <110>
Hole
Mobili
ty (cm
2 /Vse
c)
(001)------- (110) [Pham] This Work
(001) x'= <100>y'= <010>
Anisotropic SR scattering
*A. T. Pham, C. Jungemann, and B. Meinerzhagen, “Microscopic modeling of hole inversion layer mobility in unstrained and uniaxially stressed Si on arbitrarily oriented substrates,” Solid-State Electronics, vol. 52, pp. 1437-1442, May 2008.
National Chiao-Tung University Nano Electronics Physics Lab. 13
; ;
Two-Dimensional Electron Gas
IEE5328 Prof. MJ Chen NCTU 14
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Under the momentum relaxation time approximation,
Thick Oxides
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Thick Oxides
Gaussian model:
Exponential model:
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Thin Oxides
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Thin Oxides
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20
Inversion layer mobility in thick oxide MOSFETs can be limited to three primary scattering mechanisms:
Total Mobility
Effec
tive
Mobility
Effective Field
Coulmbscattering phonon
scattering
Surface roughnessscattering
Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region- The scattering rate of ionized impurity in 3-D case can be presented by:
- However, it is not the 2-D electron gas inside the MOSFET.
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Ionized Impurity Ionized Impurity Scattering ModelScattering Model
: the ionized impurity concentration
: the Debye length can be written as , where n0 is the 3-D density of the mobile carrier
34 222 2 2 22
2 22 2
81[ln(1 ) ] , 4 ( )
( ) 116 2I D
Dimp o si
N e mELr pr E r L
E rm
IN
DL 20
si o BD
k TL
e n
Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region at 2-D case- The momentum conservation in the z-direction of the 3-D case at the scattering process of 2-D carriers should be replaced by the integral as:
Therefore, the scattering rate of ionized impurity scattering in 2-D case
from mth subband to nth subband can be expressed as:
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where H2D and H3D are the matrix elements for 2-D and 3-D scattering, respectively.
22 22 3 ( ) ( ) ( ) ( ) ( ), ziq z
D D z z z mn z m nH H I q dq I q I q z z e dz
34 2 1
2 2 222,, 22 2
, 2 / 4 3 ,
( )1 1[ln(1 ) ] , ( ) ( )
( ) 1 ( )16 2I D
m n m nm nimp D m no si
N e g Err E W z z dz
E r g E Wm
22 0
2 2
8, ox av BD
Dinv
Z k TmELr L
e N
• :The average inversion layer thickness
• :the density of states for two dimensions • :the density of states for three dimensions
avZ2 ( )Dg E
3 ( )Dg E
Ionized Impurity Ionized Impurity Scattering ModelScattering Model
- For intravalley phonon scattering model, the momentum-relaxation rate in the subband mth to nth:
The total scattering rate in mth subband is determined by summing up within all the subbands:
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Phonon Scattering ModelPhonon Scattering Model
21
(2 / 4) (2 / 4) 2 2,2, 3
(2 / 4) ,
1 1, ( ) ( )
acv d ac B
m n m nm nac m nl
n m D k TW z z dz
Ws
Dac : deformation potential due to acoustic phononsSl : sound velocity ρ : crystal densityWm,n: the form factor determined by the wave-functions of the mth subband and nth subbands
,(2 / 4) (2 / 4)
( )1
( ) ( )m
m m nnac ac
U E E
E E
where ( ) 1( 0) 0( 0)U x x and x is a step function.
- For the intervalley phonon scattering model: (Incorrect Version) (1). From mth subband in twofold valleys to the nth subband in fourfold valleys:
(2). From mth subband in fourfold valleys to the nth subband in twofold valleys:
(3). From mth subband in fourfold valleys to the nth subband in fourfold valleys:
(4). From mth subband in twofold valleys to the nth subband in twofold valleys :
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Phonon Scattering ModelPhonon Scattering Model
,
2{ } 14 ' 2 ' 2
, '2 ,
4 1 ( )1 1 1 1( ), ( ) ( )
( ) 2 2 2 1 ( ) m n
fd k k
k k n m nm nkINTER k m n
m D f E EN U E E E W z z dz
E E W f E
2{ } 1
2 ' ' 2 2,,
4 ,
2 1 ( )1 1 1 1( ), ( ) ( )
'( ) 2 2 2 1 ( )
fd k k
k k n m n m nm nkINTER k m n
m D f E EN U E E E W z z dz
E E W f E
2{ }4 '
, '4 ,
2{ }4 ' 1
',
2 1 ( )1 1 1 1( )
( ) 2 2 2 1 ( )
1 ( )1 1 1( ) , (exp( ) 1)
2 2 2 1 ( )
fd k k
k k nm nkINTER k m n
gd k k k
k k n kk k m n B
m D f E EN U E E E
E E W f E
m D f E E EN U E E E N
E W f E k T
where Ek and Dk are deformation energy and potential at kth intervalley phonon, and Nk is the occupation number of kth intervalley phonon.
2{ }2 ' 1
, '2 ,
1 ( )1 1 1 1( ) , (exp( ) 1)
( ) 2 2 2 1 ( )
fd k k k
k k n km nkINTER k m n B
m D f E E EN U E E E N
E E W f E k T
- The scattering rate for a Gaussian function is described as:
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Surface Roughness Scattering Surface Roughness Scattering ModelModel
2 2( ) 2 2 2 2 2
4, 3
0
( )1( ) (1 cos )
( ) 2DOS eff
j ij q
ji jSR
m E e EU E E e d
E
2 2 2 2=2 (1 cos ) =4 sin2
q k k
( )2
2
2 (E-E ) = DOS
jjm
k
Δ : rms height of the amplitude of surface roughness
: correlation length of surface roughness
0( ) ( )
eff
ij j idVE z z dz
dz
Derivation of Two-Dimensional Mobility in the Universal Mobility Region- The scattering rates of the twofold and fourfold valley:
The electron mobility by using the average energy within the 2DEG in the relaxation time approximation can be given as
The total universal mobility averaged over the subband occupation is described by
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Electron Mobility ModelElectron Mobility Model
2 2 2 4 4 4
1 1 1 1 1 1,
( ) ( ) ( ) ( ) ( ) ( )i i i i i iphonon SR phonon SRE E E E E E
'
'2 4
2 4'
2 4
( ) ( )( ) ( ) ( )( ),
( )( ) ( )( )
i i
i i
i ii iE Ei i
c i c iE E
f fe E E E dE e E E E dE
E Ef f
m E E dE m E E dEE E
'2 4 '
'
( )i ii i
i iuni
s
N N
N
The physical parameters for phonon and surface-roughness
electron mobility used for Si in this work
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Electron Mobility ModelElectron Mobility Model
The Universal mobility Curve - Universal electron mobility includes phonon scattering and surface roughness
scattering, which is independent of process parameters, especially when plotted versus of high effective field (Eeff).
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Electron Mobility ModelElectron Mobility Model
0
( ), 0.5inv dep
effsi
e N NE
• : total inversion layer charge densityinvN
• :the surface concentration of the depletion charge
depN
10-1 100101
102
103
104
Dk=11.5x108 eV/cm Dac=13 eV
=14.9x10-8 cm =2.9x10
-8 cm
Eff
ective
Mobility
(cm
2 / V
s)
Eeff
(MV/cm)
Nsub
=7.7X1017cm-3
Nsub
=3.0X1017cm
-3 N
sub=7.2X10
16cm-3
Nsub
=2.0X1016cm-3 N
sub=3.9X10
15cm-3
Dots:Exp.(Takagi,et al.[10])Lines:Sim.(This work)
Npoly
=1x1020cm-3
T=300KT
ox=25nm
0.1 1102
103
104
Lines:Sim. total mobility cosists of uni
&imp
(This work)
T=397 K T=347 K T=297 K T=242 K
Exp. T(K) 397
347
297
242
Nsub
=3.9X1015cm
-3
T=397 K T=347 K T=297 K T=242 K
Mobili
ty (c
m2/V
s)
Eeff
(MV/cm)
=14.9x10-8 cm =2.9x10
-8 cm
Dk=11.5x108 eV/cm Dac=13 eV
Dots:Exp. data (Takagi, et al [10] )
Electron Effective Mobility:Thick Oxides
IEE5328 Prof. MJ Chen NCTU 29
Current NEP-electron-mobility simulator was developed under the parabolic band approximation, the isotropic scattering approximation, the elastic scattering approximation (surface roughness and impurity scattering), and the momentum relaxation approximation.
30
Ionized Impurity
Scattering
2
4D
r
Ls
si
qU e
r
2si B
D
k TL
q n
34 2 2 222
1[ln(1 ) ]
( ) 116 2I
imp si
N qE
E m
2
22
8 DmEL
The perturbing potential is the screened Coulomb potential:
r: the distance from the scattering centerLD: Debye length
n: 3-D density of the mobile carriers and equal to Ninv/Zav
Through the Fermi’s Golden Rule, the relaxation time due to ionized impurity
scattering is:
NI: 3-D impurity concentration, which is about equal to
Nsub
Here, q means the elementary charge.
31
Impurity Scattering
The momentum relaxation time for scattering of 2-D carriers from uth subband to vth subband:
34 22 2 222
, 22, 2/4 3
1 ( )[ln(1 ) ] ( ) ( )
( ) 1 ( )16 2I D
u vu vimp Dsi
N q g EE z z dz
E g Em
P.S. Only consider intra-subband
K. Hirakawa and H. Sakaki, “Mobility of the two-dimensional electron gas at selectively doped n-type AlxGa1-xAs/GaAs heterojunctions with controlled electron concentrations,” Phys. Rev. B, Condens. Matter, vol. 33, no. 12, 8291-8303, Jun. 1986.
M. Lundstrom, “Fundamentals of carrier transport,” Cambridge University Press, 2000.
32
Phonon Scattering
1. intravalley phonon scattering model: (acoustic phonon) the momentum-relaxation time for scattering from the uth subband to the vth subband
2, 2/4 2 2 1
, ( 2/4) ( 2/4), ( 2/4),, 3 2int ( 2/4) , ( 2/4)
1 1( ) , ( ( ) ( ) )
( )d ac B
v u v u vu vra l u v
m D k TU E E W z z dz
E s W
kB : Boltzmann constant , Dac : deformation potential due to acoustic phonons ρ : crystal density , Sι: sound velocityU(x): step function
S. Takagi, J. L. Hoyt, J. J. Welser, and J. F. Gibbons, “Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal-oxide-semiconductor field-effect transistors,” J. Appl. Phys., vol. 80, no. 3, pp. 1567-1577, Aug. 1996.
D.Esseni, A. Abramo, L. Selmi, and E. Sangiorgi, “Physically Based modeling of Low Field Electron Mobility in Ultrathin Single- and Double-Gate SOI n-MOSFETs”, IEEE Trans. Electron Devices, vol. 50, no.12, pp.2445-2455, 2003.
K. Uchida, A. Kinoshita, and M. Saitoh, “Carrier transport in (110) nMOSFETs: Subband structures, non parabolicity, mobility characteristics, and uniaxial stress engineering,” in IEDM Tech. Dig., pp.1019-1021, 2006.
33
Phonon Scattering model (Correct Version)
2. intervalley phonon scattering model: (optical phonon) (a) From uth subband in twofold valleys to the vth subband in twofold valleys:
(b) From uth subband in twofold valleys to the vth subband in fourfold valleys:
(c)From uth subband in fourfold valleys to the vth subband in twofold valleys:
(d)From uth subband in fourfold valleys to the vth subband in fourfold valleys:
,
2{ } 1,, 2 ' 2 2
, , 2, 2, 2,, 'int , 2 2 , ,
1 ( )1 1 1 1( ) ( ), ( ) ( )
( ) 2 2 2 1 ( ) u v
gk gd k
k g k g v u vu vker k g u v
f E Em DN U E E E W z z dz
E E W f E
2{ },, 4 2 2 1
, , 4, , 2, 4,,int , 2 4 , ,
1 ( )41 1 1 1( ) ( ), ( ( ) ( ) )
( ) 2 2 2 1 ( )
fk fd k
k f k f v u v u vu vker k f u v
f E Em DN U E E E W z z dz
E E W f E
2{ },, 2 2 2 1
, , 2, , 4, 2,,int , 4 2 , ,
1 ( )21 1 1 1( ) ( ), ( ( ) ( ) )
( ) 2 2 2 1 ( )
fk fd k
k f k f v u v u vu vker k f u v
f E Em DN U E E E V z z dz
E E V f E
2{ },, 4
, , 4,, 'int , 4 4 , ,
2{ },, 4 ' 2 2 1
, , 4, , 4, 4,', ,
1 ( )21 1 1 1( ) ( )
( ) 2 2 2 1 ( )
1 ( )1 1 1( ) ( ), ( ( ) ( ) )
2 2 2 1 ( )
fk fd k
k f k f vu vker k f u v
gk gd k
k g k g v u v u vk k g u v
f E Em DN U E E E
E E V f E
f E Em DN U E E E V z z dz
E V f E
34
2
4
4 4
4
f-type
f-type
f-typef-
typ
e f-typ
ef-type
f-typef-typ
e
Acoustic phonon
Acoustic phonon
Acoustic phonon
Acoustic phononAcoustic phonon
g-type
g-type
g-type
35
Surface Roughness Scattering2 22 2 2 2 2
4, 3
0
2 2 22
( )1( ) (1 cos )
( ) 2
2 ( )where 2 1 cos and
i qDOS eff
ii iSR
DOS j
m E e EU E E e d
E
m E Eq k ( θ) k
0( ) ( )i i i
eff
dVE z z dz
dz
ki
kjq
E1
E2
P.S. Only consider intra-subband
Yamakawa, H. Ueno, K. Taniguchi, C. Hamaguchi, K. Miyatsuji, K. Masaki, and U. Ravaioli, “Study of interface roughness dependence of electron mobility in Si inversion layers using the Monte Carlo method,” J. Appl. Phys., vol. 79, no. 2, pp. 911-916, Jan. 1996.
D.Esseni, “On the Modeling of Surface Roughness Limited Mobility in SOI MOSFETs and its Correlation to the Transistor Effective Field”, IEEE TED, Vol.51 NO.3, pp.394-401, 2004.
36
Three-Dimensional Stress Effect on Electron Mobility
-3000 -1500 0 1500 30000
1
2
(001)/<110> nMOSFETSimulation of DG-NEP at E
eff=1MV/cm
Long./Tran. Vert. Biax.
Mobility R
atio
Stress (MPa)-3000 -1500 0 1500 30000
1
2
3
4
5M
obility R
atio
Stress (MPa)
(110)/<1-10> nMOSFETSimulation of DG-NEP at E
eff=1MV/cm
Long./Vert. Tran. Biax.
-3000 -1500 0 1500 30000
1
2
3
Mobility R
atio
Stress (MPa)
(111)/<1-10> nMOSFETSimulation of DG-NEP at E
eff=1MV/cm
Long. Tran. Biax. Vert.